Pál Turán

Pál Turán: Theorems From a Labour Camp

In July 1944, a Hungarian Jew in forced labour was pushing cars of bricks along rails at a factory near Budapest. Where the tracks crossed, the cars jolted and the work got harder. Most men would have cursed the crossings. Turán began wondering what the minimum possible number of crossings was for m kilns and n warehouses — and in doing so posed what is now called Turán's brick factory problem, one of the founding questions of the theory of crossing numbers. He was a prisoner. He was also, at that moment, doing mathematics.

Budapest, Fejér, and a Closed Door

Turán was born on 18 August 1910 in Budapest, then Austria-Hungary, into a Hungarian Jewish family. His talent showed in secondary school. He took his teaching degree from the University of Budapest in 1933 and his PhD in 1935 at Eötvös Loránd University under Lipót Fejér, the central figure of Hungarian analysis.

He published two significant papers in 1933, in the journals of the American and London Mathematical Societies. It made no difference to his employability. Hungary's *numerus clausus* laws restricting Jewish participation in higher education kept him out of stable academic work for years, and the position he eventually secured in 1938 was at a rabbinical training school in Budapest. Here is a mathematician publishing internationally at twenty-three and teaching at a seminary at twenty-eight because the state had legislated him out of the university.

Meeting Erdős

On 1 September 1930, at a mathematical seminar at the University of Budapest, Turán met Paul Erdős. Both had been noted answerers in the problem journal KöMaL as schoolboys. The collaboration that began that day lasted forty-six years and produced twenty-eight joint papers — one of the great sustained partnerships in twentieth-century mathematics, and the source of much of what we know about Turán's character, because Erdős wrote about him at length and did not flatter.

Five Years in the Camps

In September 1940 Turán was arrested on account of his Jewish heritage and sent to labour camps. He spent five years in various camps across Transylvania and Hungary.

The period should have been a void in his bibliography. It was not. An officer named Joshef Winkler, himself trained in mathematics, recognised Turán's name and arranged lighter duties for him. Turán used the margin this bought him. He composed a significant paper on the Riemann zeta function while interned, and kept himself intact by solving problems in his head and thinking through problems. The brick factory observation came in July 1944; he took it up seriously only after 1952, but the question was born on the rails.

Erdős put it flatly: Turán "did not lose his spirit even in the Nazi camps and did brilliant work there." He was liberated in 1944 and returned to teaching.

Extremal Graph Theory

The most consequential thing Turán built, he built while imprisoned. Erdős again: "In 1940–1941 he created the area of extremal problems in graph theory which is now one of the fastest-growing subjects in combinatorics."

The founding result, Turán's theorem, gives an upper bound on the number of edges a graph can have without containing a complete subgraph on r vertices. To prove it he invented the Turán graph, a generalisation of complete bipartite graphs. The theorem's importance is not the specific bound but the question form it established: how much of a structure can you have before some pattern is forced to appear? That question, asked across thousands of settings since, is what extremal combinatorics is. The Kővári–Sós–Turán theorem extended the approach to bipartite graphs with forbidden subgraphs.

The Sieve and the Power Sum

Number theory was his primary field. The Turán sieve, from 1934, gave a simplified proof of a 1917 result of Hardy and Ramanujan on the normal order of the number of distinct prime divisors. Halász identified its real significance: "Its true significance lies in the fact that it was the starting point of probabilistic number theory." A simplification that founds a discipline is not a simplification. The Turán–Kubilius inequality generalised the work.

He coined the term "prime number race" for the irregularities in how primes distribute across residue classes, working with Knapowski on results bearing on Chebyshev's bias. The Erdős–Turán conjecture concerns primes in arithmetic progression. And he developed the power sum method, a technique for inequalities on sums of powers of the zeros of the Riemann zeta function, with applications reaching into complex analysis, numerical analysis, differential equations, transcendental number theory and the estimation of zeros generally.

On the hypothesis itself, Erdős recorded that Turán was "an 'unbeliever,' in fact, a 'pagan': he did not believe in the truth of Riemann's hypothesis." His analysis produced Turán's inequalities on Legendre polynomials and, with Erdős, the Erdős–Turán equidistribution inequality.

A Diagnosis Kept Secret

He became associate professor at the University of Budapest in 1945 and full professor in 1949. The Kossuth Prize came in 1948 and again in 1952; he was elected corresponding member of the Hungarian Academy of Sciences in 1948 and ordinary member in 1953. He founded and presided over the János Bolyai Mathematical Society, sat on the editorial boards of leading journals, spoke at the International Congress of Mathematicians in 1970 and was invited to the Fields Prize committee that year. The Tibor Szele Prize followed in 1975.

He married Edit Kóbor in 1939, with whom he had a son, Róbert; in 1952 he married the mathematician Vera Sós, with whom he published jointly and had two sons, György and Tamás.

Around 1970 he was diagnosed with leukaemia. Only Vera knew. She chose not to tell him, believing he was "too much in love with life" and would have despaired. She told Erdős only in 1976. Erdős later lamented the cost: not knowing his time was short, Turán postponed certain works "for later" and never completed them.

Why Pál Is Called a Genius

The specific faculty is problem-creation at the level of whole disciplines. Turán's individual theorems are strong, but his signature is that three separate areas trace their origin to him: extremal graph theory, which Erdős explicitly credits him with creating in 1940–41; probabilistic number theory, which Halász says began with his 1934 sieve; and the theory of crossing numbers, which began with bricks. Founding one field is a career. Founding three is something else — an unusual instinct for the question underneath the question, the general form of which a specific difficulty is an instance.

The circumstances sharpen the claim rather than merely decorating it. He created extremal graph theory in a labour camp, with no library, no colleagues and no paper, holding the structures in his head. Erdős, who was not sentimental about mathematical ability and knew everyone worth knowing, said he did brilliant work there.

The counter-case is thinner than usual but real. Erdős was his closest friend for forty-six years and his most quoted assessor, which is not an independent source. Turán's power sum method, for all its reach, did not crack the Riemann hypothesis he disbelieved. His honours — two Kossuth Prizes, the Szele Prize, academy membership — are Hungarian national recognitions rather than the discipline's top international awards. And extremal graph theory's explosive growth owes much to those who developed it after him. He set the question; others built the subject.

Legacy

Turán died of leukaemia in Budapest on 26 September 1976, aged sixty-six, still unaware of what was killing him. Every combinatorics course now teaches his theorem, and probabilistic number theory dates its founding to a sieve he devised at twenty-four. The brick factory problem remains open in general. It is a fitting monument: a question posed by a prisoner watching rails cross, which the discipline has not yet finished answering.